Lowering Mechanical Noise Floor in Speakers - page 3
Once the TL values have been calculated, assessing the effective reduction of transiting acoustical energy by the panel becomes a simple matter of subtracting TL from the dB SPL level of the source, in this case, those levels obtaining in the immediate vicinity of the cabinet-internal side of the panel.
These expressions are, of course, panel-focused and the majority of transmission loss theory found in the literature - upon which these equations are based - is for wall-sized (or larger) "panels" dividing room-sized "enclosures" sporting ideal (or completely lacking) boundary conditions. The results are not necessarily applicable to panels of the material type, dimensions and boundary conditions commonly found obtaining in loudspeaker enclosures.
A search for a better enclosure-focused approach to characterizing the outward re-radiation of acoustic energy by an enclosure's panels brings us to the concept of insertion loss. Mathematically, the general expression of insertion loss (IL) is defined as:
Where:
= acoustic sound pressure level, external to cabinet panel
= acoustic sound pressure level, internal to cabinet panel
The equation is instructive but nevertheless descriptive in nature.
Fortunately, there exists a predictive mathematical expression for determining the insertion loss of a loudspeaker cabinet panel that takes in to account various factors such as enclosure volume impedance, panel mechanical impedance and so forth. For a close fitting cabinet (one where the distance from internal sound source to any panel is 1m or less), with clamped boundary conditions obtaining, IL can be mathematically stated by:
Where:
k = acoustic wave number, 
λ = wavelength
d = distance from source to panel
density of air
c = speed of sound in air
and
Where:
a = panel length
b = panel width
h = panel thickness
ρ = density of panel material
ω = frequency of interest
and
which is the complex bulk modulus of the panel material
η = internal damping coefficient of the panel material
E = Young's modulus or modulus of elasticity
ν = Poisson's ratio (ratio of orthogonal straining)
For a polymer composite panel with the material properties as listed in the above table and with the dimensions: a = 20" (.508 m), b = 10" (.254 m), and h = .75" (.01905 m) we can calculate 
, K and finally IL.
Graphing IL we get:
Here we see a series of nulls alternating between peaks in IL values. Clearly, the response of a cabinet's panels are dependent on both the frequency of the various excited modes and the frequency generated by the driver enclosed within the cabinet; the IL response at higher frequencies is controlled by various material properties along with the distance between the panel and driver.
When panel vibration is in phase with the incident pressure wave, the panel becomes acoustically "invisible" to the incident pressure wave. Standing waves within the enclosure, having the characteristic values of:
, 
, 
and frequencies that can be approximated by:
Where:
c = speed of sound in air,
nx, ny, nz = 0, 1, 2, 3 …
lx, ly, lz = internal length, width depth
can produce acoustic wave pressures greater than that generated by the driver itself. This can result in panel vibration of a magnitude sufficient to be considered a sort of negative insertion loss.
Curious to see how varying the values of various parameters would affect the IL dB levels, I ran a detailed series of simulations.
First I varied panel length.
Then I varied panel width
Varying length and width, individually, did indeed alter the IL response spectrum. I then varied a and b, simultaneously. In all three cases, as the panel became smaller we see an increase in the frequency of the first null as well as an increase in IL
I then varied panel density. Across the IL response spectrum we see that increasing density increased the amount of insertion loss. All else being equal, we can thus expect that the denser a given panel's material, the quieter it will be.
Then, I varied panel thickness. Increasing panel thickness increases effective stiffness, which results in increased insertion loss, particularly noticeable at the lower end of the IL response spectrum, where insertion loss is controlled directly by the panel's effective stiffness.
